The lack of signature by the other party is probably unimportant, particularly since the parties apparently operated under the agreement for a period of time, and the party who didn't sign (you really don't know that the company didn't sign, only that it didn't provide your son with a copy of the signed agreement) and the nonsigning party delivered a copy of the agreement to your son, but that doesn't mean that the non-compete provision is enforceable.
Unless the parties specifically contemplated that they would not have an agreement until both parties signed (the fact that your son worked for a while pursuant to the contract indicates that this was not such a case), any court will probably only be concerned that the contract was signed by the party against whom it is sought to be enforced, i.e. your son, if he is sued for breaching the non-compete clause (known as a restrictive covenant).
However, restrictive covenants in employment (although your son may have been an independent contractor, I think his situation would be analyzed the same way as an employee's) are somewhat disfavored (in California they are virtually verboten, but other states are more open to them) and generally will only be enforced to the extent necessary to protect the employer's vital interests. There are many factors which factor into the analysis of whether a restrictive covenant will be enforced, including whether the employee quit or was fired, whether the employee has any special skills or training, whether the employer spent any time or money training the employee, how long the covenant runs and how wide is the geograpical area it covers, whether there are any trade secrets or confidential information of the employer which are being protected, whether the employee received any additional compensation for entering into the non-compete agreement, whether the employer enforces the such coveants uniformly or selectively, the type of industry in which the parties operate, etc., etc. My gut feeling from your brief description of your son's job is that this covenant would not be enforced.
However, even if a restrictive covenant is not enforceable on its face, courts will sometimes "blue pencil" a clause to make it enforceable. In other words, if a court deems the 1 1/2 year restriction to be too long, or the geographical scope of the clause too wide, but otherwise thinks the covenant is reasonable, it may modify the restriction to only last for a shorter period or cover a smaller geographical area....
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